Group 4 C4 t: f& h( z- iA group is defined as a finite or infinite set of Operands& h( @1 R* A1 H m1 z
(called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator ; E8 K$ W- x$ Q- U6 { to form well-defined products and which furthermore satisfy the following conditions: j1 L' O" R4 B1. Closure: If and are two elements in , then the product is also in . # K6 V5 _& q9 l) S
2. Associativity: The defined multiplication is associative, i.e., for all , . 1 U$ I& [3 E! S! N3. Identity: There is an Identity Element5 \5 n/ r6 g D t c' O
(a.k.a. , , or ) such that for every element . 2 u" @5 `* x0 M8 y/ n0 ?- ]
4. Inverse: There must be an inverse or reciprocal of each element. Therefore, the set must contain an element such that for each element of . ; a+ S7 Y/ q1 I' j6 ]0 @+ ] `
A group is therefore a Monoid $ q' E) r! y' ~4 \1 @5 o for which every element is invertible. A group must contain at least one element. 0 R- F z+ n' R1 m1 o
9 H/ A8 J# T" z0 l2 J! ~7 B" }The study of groups is known as Group Theory, k/ F6 V1 F5 P1 l
. If there are a finite number of elements, the group is called a Finite Group' i V+ s. K$ L# _) G& o) d6 A
and the number of elements is called the Order3 S- V& }% C- C$ ]2 s9 ?
of the group. ! Z7 k% B2 k& |; n; p+ W$ x" m
+ D; x0 m6 d: r6 {! T
Since each element , , , ..., , and is a member of the group, group property 1 requires that the product . n3 d. Q" t/ K6 ^& z0 W# Z
6 }; D p8 L. H( N0 X5 j+ [ E
(1); i, e9 Z' k, i. M
; o2 R2 k6 n' A( {0 ~# {4 D
8 I4 W3 u& M, m8 h, [: c& v
- X" _* y9 f" w' \# M
must also be a member. Now apply to , 5 p- e5 @! Y: o) a% m( x ( ~ \5 l& J8 Z3 S, y
+ N5 J$ m' X2 W3 W
(2) ( B) N0 n1 O, P7 R4 `
0 i0 E; I% Y$ q0 u- W- w2 I ! v8 I: K, C$ S2 B; P " U, `' |, @& OBut ) t$ M" e- L u5 n5 j$ Y( r5 X
, k3 H& ^$ I5 H$ ~# U5 H: X2 g
# ~5 v' d8 l; s" ?. ^3 l, }
7 E- |- J# E- p0 v' p& y
. |/ t; Y. f3 U* E
& y6 x0 m! K) \
(3) 4 Z. x, E: j: y& L, A, \
so - @0 a/ B6 D' d5 l2 j
% J- \+ ^7 N5 o. U! @9 k7 v: u2 {
(4) ; f/ H' k* u6 F) H( q& t
. R- B. I" Y( W3 r3 V d 9 j! p9 i( v8 }, U; C! w- J! ]. J1 _/ g7 _' a
which means that 9 G z" J, x/ @. T
* K' i2 `/ N! E( N: X# c) U% [/ B+ c
(5). Y N! ]( w! o; w- Z
4 y5 M5 ^3 c( m* T 4 t2 b! B! Q! h2 v, S# J$ C8 h ~# Z! [, V; i. m1 uand 7 x$ k( m. G: ]5 x3 i: A6 s9 f
; @% j- c3 B7 a4 ~" R3 k2 g
(6)8 ^6 a$ P# _1 R( j" Y2 e
2 u9 a9 V, I, h2 w/ ~4 F$ q3 S' {
5 u; g; s' w6 k6 g, Z " \% q$ Z( S. [& e/ g5 M* y3 Y 5 q1 U: W, p4 l( b